Motion Equations of Linear Systems

This section provides a quick introduction of motion equations of linear systems, which are first order linear differential equations of the canonical coordinates.

What Is Linear System? A Linear System is a system where motion equations are first order linear differential equations of the canonical coordinates. In this case, motion equations can be expressed in the vector form as shown below:

```x' = Ax + b                            (P.12)
where:
x is the vector of [q1, q2, ..., p1, p2, ....]
x' is the vector of dx/dt
A is a matrix of coefficients
b is a constant vector
```

For a Linear System of a single object with 1 degree of freedom, vector x is [q, p] and x' is [q', p']. The motion equations can be expressed as:

```  |q'| = |a11, a12| . |q| + |b1|
|p'|   |a21, a22|   |p|   |b2|
```

Now let's look at those systems of motions presented earlier:

1. The Free Fall Motion of a single object is a linear system. The motion equations can be expressed as:

```  m*g = -p'                             (P.4)
p/m = q'                              (P.5)

In vector form:
|q'| = |0, 1/m| . |q| + |   0|
|p'|   |0,   0|   |p|   |-m*g|
```

2. An object on a spring moving horizontally is a linear system. The motion equations can be expressed as:

```  kq = -p'                              (P.7)
p/m = q'                              (P.8)

In vector form:
|q'| = | 0, 1/m| . |q| + |0|
|p'|   |-k,   0|   |p|   |0|
```

3. A simple pendulum is not a linear system, because p' is not linearly related to q:

```  m*g*l*sin(q) = -p'                   (P.10)
p/(m*l*l) = q'                       (P.11)
```